Two people with the same birthday
You and a friend are standing outside a room containing 25 people, all of whom are strangers to both of you. He says, "I will give you $20 if there are not two people in that room who share a birthday (month and day). Otherwise, if there are, you pay me $20."
Do you take that bet? (In other words, is there a greater than or less than 50% probability that two people in the room share a birthday?)
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1 solution
In a room of 25 people, there are 2 2 5 • 2 4 = 2 6 0 0 = 3 0 0 unique pairs of people. (Want to change your answer yet?)
Furthermore, there is a 3 6 5 3 6 4 probability that a given pair of people have different birthdays.
Therefore, the probability that all people in the room have different birthdays is ( 3 6 5 3 6 4 ) 3 0 0 , which evaluates to about .439.
In a room containing 25 people, the likelihood that there are not 2 people who share a birthday is 43.9%. You would not be wise to accept this bet.